That's not really much of a problem, but still a nice stumbling block for inexperienced and an example of the evilness of formal symbol manipulations, even innocent-looking ones.
Consider a function $f:(x,y)\to \mathbb{R}$ and a change of coordinates $(x,y)\mapsto (x,xy)$. Find partial derivatives in this new chart. If one is acting formally, one can assume $$\left(\frac{\partial f}{\partial x}\right)_{new} = \left(\frac{\partial f}{\partial x}\right)_{old}$$ but then the change-of-chart formula gives $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial (xy)}$$ so $\frac{\partial f}{\partial (xy)} = 0$ - an obvious contradiction. This is a good example of abuse of notation leading to fallacy and a reminder that partial derivatives are taken not with respect to a lone coordinate, but with respect to a chart, a vector field in a general case. Also an example of the relative nature of coordinates.
